Chapter 1

Game TheoryFor Microeconomics at MPP Chapter 1: Static Games of Complete Information... Definition The normal-form representation of an n-player game specifies the player’s strategy spac

Game Theory

For Microeconomics at MPP

Chapter 1: Static Games of Complete Information

Normal-form Games

The Prisoner’s Dilemma game

Deviation Cooperation

Cooperation Deviation

Prisoner 1

Prisoner 2

－ 1 －－ 1

－ 6 －－ 6

0 － － 9

－ 9 － 0

Definition The normal-form representation of an n-player

game specifies the player’s strategy spaces, S1,…,Sn and their payoff functions u1,…,un We denote this game by G={ S1, …,Sn ;

u1,…,un }

Strictly Dominated Strategies

Definition Strategy s’i is strictly dominated by strategy s’’i if for each

combination of other player’s strategies, i’s payoff from playing s’i is strictly less than i’s payoff from playing s’’i

Iterated Elimination of Strictly Dominated

Strategies

Up

Down

Player 1

Player 2

1 － 2 0 － 1

0 － 3

1 － 0

0 － 1 2 － 0

T M B

4 － 0

0 － 4

3 － 5

4 － 0

0 － 4

3 － 5

5 － 3

5 － 3

6 － 6

Nash Equilibrium

Definition In the normal-form game G={ S1, …,Sn ; u1,…,un }, the strategies

(s*1,s*2,…,s*n) are Nash equilibrium if for each player i, s*i is player i’s best

response to the strategies specified for the n-1 players, (s*1,…,s*i-1,s*i+1,

…,s*n)

T M B

0 － 4 4 － 0 5 － 3

4 － 0 0 － 4 5 － 3

3 － 5 3 － 5 6 － 6

Opera Opera

Fight

Fight

2 － 1

1 － 2

0 － 0

0 － 0

Pat

Chris

The Battle of the Sexes

Mixed Strategies and Existence of Equilibrium

Player 2

Player 1

Heads

Tails

Heads Tails

-1 － 1

1 － -1 -1 － 1

1 － -1

Matching Pennies

Player 2

Hawkish

Dovelike

Hawkish Dovelike

-1 － -1

0 － 2

2 － 0

1 － 1

Hawk- Dove game

Definition In the normal-form game G={ S1, …,Sn ; u1,…,un }, suppose

Si={si1,…,siK} Then a mixed strategy for player i is a probability distribution

pi=(pi1,…piK), where 0≤piK≤1 for k=1,…,K and pi1+ －－－ +piK=1.

Player 1

Player 2

T

M

B

3 －－

0 －－

1 －－

0 －－

3 －－

1 －－

A mixed strategy strictly dominates B

T

M

B

Player 1

Player 2

3 －－ 0 －－

0 －－ 3 －－

2 －－ 2 －－

B is best response for player 1 to some mixed strategy of 2, (q,1-q).

Existence of Nash Equilibrium

Theorem (Nash (1950)) In the n-player normal-form game G={ S1, …,Sn ; u1,

…,un }, if n is finite and Si is finite for every i, then there exists at least one Nash

equilibrium, possibly involving mixed strategies

For any strategic (or social) situation, there is at least one equilibrium.

However, multiple equilibria are probable

A useful property of mixed-strategy Nash

equilibria

Each strategy in the support of a mixed Nash equilibrium strategy earns the same payoff for the other players’ mixed Nash equilibrium strategy

Given a mixed-strategy pi, the support of pi is the set

{sij Si | pij>0}, i.e., the set of strategies assigned with

positive probability